let n be non empty Nat; :: thesis: for S being non empty non void n PC-correct PCLangSignature

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds (A \imp B) \imp (\not (A \and (\not B))) in F

let S be non empty non void n PC-correct PCLangSignature ; :: thesis: for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds (A \imp B) \imp (\not (A \and (\not B))) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B being Formula of L holds (A \imp B) \imp (\not (A \and (\not B))) in F

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds (A \imp B) \imp (\not (A \and (\not B))) in F

let A, B be Formula of L; :: thesis: (A \imp B) \imp (\not (A \and (\not B))) in F

A1: (A \imp B) \imp ((\not A) \or B) in F by Th82;

( (\not A) \imp (\not A) in F & B \imp (\not (\not B)) in F ) by Th64, Th34;

then ((\not A) \or B) \imp ((\not A) \or (\not (\not B))) in F by Th59;

then A2: (A \imp B) \imp ((\not A) \or (\not (\not B))) in F by A1, Th45;

((\not A) \or (\not (\not B))) \imp (\not (A \and (\not B))) in F by Th73;

hence (A \imp B) \imp (\not (A \and (\not B))) in F by A2, Th45; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds (A \imp B) \imp (\not (A \and (\not B))) in F

let S be non empty non void n PC-correct PCLangSignature ; :: thesis: for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds (A \imp B) \imp (\not (A \and (\not B))) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B being Formula of L holds (A \imp B) \imp (\not (A \and (\not B))) in F

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds (A \imp B) \imp (\not (A \and (\not B))) in F

let A, B be Formula of L; :: thesis: (A \imp B) \imp (\not (A \and (\not B))) in F

A1: (A \imp B) \imp ((\not A) \or B) in F by Th82;

( (\not A) \imp (\not A) in F & B \imp (\not (\not B)) in F ) by Th64, Th34;

then ((\not A) \or B) \imp ((\not A) \or (\not (\not B))) in F by Th59;

then A2: (A \imp B) \imp ((\not A) \or (\not (\not B))) in F by A1, Th45;

((\not A) \or (\not (\not B))) \imp (\not (A \and (\not B))) in F by Th73;

hence (A \imp B) \imp (\not (A \and (\not B))) in F by A2, Th45; :: thesis: verum